Whakaoti i te {l}{x+y=78}{2x+4y=200}

$$\left. \begin{array} { l } { x + y = 78 } \\ { 2 x + 4 y = 200 } \end{array} \right.$$

Whakaoti mō x, y

$$x=56$$

$$y=22$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$x+y=78,2x+4y=200$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$x+y=78$$

Me tango y mai i ngā taha e rua o te whārite.

$$x=-y+78$$

Whakakapia te -y+78 mō te x ki tērā atu whārite, 2x+4y=200.

$$2\left(-y+78\right)+4y=200$$

Whakareatia 2 ki te -y+78.

$$-2y+156+4y=200$$

Tāpiri -2y ki te 4y.

$$2y+156=200$$

Me tango 156 mai i ngā taha e rua o te whārite.

$$2y=44$$

Whakawehea ngā taha e rua ki te 2.

$$y=22$$

Whakaurua te 22 mō y ki x=-y+78. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-22+78$$

Tāpiri 78 ki te -22.

$$x=56$$

Kua oti te pūnaha te whakatau.

$$x=56,y=22$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$x+y=78,2x+4y=200$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}78\\200\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\2&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\2&4\end{matrix}\right))\left(\begin{matrix}1&1\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&4\end{matrix}\right))\left(\begin{matrix}78\\200\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&4\end{matrix}\right))\left(\begin{matrix}78\\200\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&4\end{matrix}\right))\left(\begin{matrix}78\\200\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-2}&-\frac{1}{4-2}\\-\frac{2}{4-2}&\frac{1}{4-2}\end{matrix}\right)\left(\begin{matrix}78\\200\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2&-\frac{1}{2}\\-1&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}78\\200\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\times 78-\frac{1}{2}\times 200\\-78+\frac{1}{2}\times 200\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}56\\22\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=56,y=22$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$x+y=78,2x+4y=200$$

Kia ōrite ai a x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2x+2y=2\times 78,2x+4y=200$$

Whakarūnātia.

$$2x+2y=156,2x+4y=200$$

Me tango 2x+4y=200 mai i 2x+2y=156 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+2y-4y=156-200$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y-4y=156-200$$

Tāpiri 2y ki te -4y.

$$-2y=156-200$$

Tāpiri 156 ki te -200.

$$-2y=-44$$

Whakawehea ngā taha e rua ki te -2.

$$y=22$$

Whakaurua te 22 mō y ki 2x+4y=200. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+4\times 22=200$$

Whakareatia 4 ki te 22.

$$2x+88=200$$

Me tango 88 mai i ngā taha e rua o te whārite.

$$2x=112$$

Whakawehea ngā taha e rua ki te 2.

$$x=56$$

Kua oti te pūnaha te whakatau.

$$x=56,y=22$$